1* 



nlr^e ) = In r + Id) 

 ^ St / St St 



to which the Cauchy-Riemann Equations have been applied. We then obtain 

 from (5) and (6) 



ZTTmCs) 



m(t) -;^ — £n r -m(s) t^ — £n r 



dn St dn St 

 s t 



dt = V (s) 



(7) 



Since the new integrand vanishes at the point s = t, where the largest con- 

 tribution to the integral in (5) occurs, the form in (7) suggests that the 

 Preston-Lighthill formula should give a good first approximation. 



From formula (3) for the source distribution, Lighthill immediately 

 concludes that the displaced surface, obtained by the addition of the dis- 

 placement thickness, is a stream surface. This again requires the assump- 

 tion that all the flux from the source distribution m(s) be outward, i.e., 

 that the given contour be an equipotential for the distribution. For then, 

 the Gauss flux theorem gives the relation 



/ 



2tt I m ds = U n(s) 

 '0 



where n(s) denotes the stream surface generated by m(s). Comparison with 

 (3) then shows that n(s) = 6^ . 



A Second Approximation for the Source Distribution - 

 Two-Dimensional Case 



It appears to be easier to derive a formula for a source distribution 

 M(x), equivalent to the BLW, for a thin, symmetric body, when M(x) is dis- 

 tributed on the axis of symmetry, the x-axis. The given profile will be 

 denoted by y = f(x), the edge of the boundary layer, EBLW, by y = g(x), and 

 the pirofile thickened by the displacement thickness by y = h(x). Put 



